Recently I learned that it can't be proved that mathematical axioms are consistent. And furthermore, in 1900s math was based on an inconsistent system of axioms.
So, are there any results, that were true in old axioms, were believed to be true (so obvious paradoxes don't count), but are false in modern axioms?
If there are some, then does it mean I can be skeptical about any proved statements, just because we can never be sure that our axioms are consistent?